NBER WORKING PAPER SERIES

ON THE OPTIMAL BURDEN OF PROOF

Louis Kaplow

Working Paper 17765http://www.nber.org/papers/w17765

NATIONAL BUREAU OF ECONOMIC RESEARCH1050 Massachusetts Avenue

Cambridge, MA 02138January 2012

I am grateful to Steven Shavell, Christopher Snyder, Kathryn Spier, the referees, and presentationparticipants at the 2011 AEA meetings, Harvard, NYU, and the NBER for comments, to the John M.Olin Center for Law, Economics, and Business at Harvard University for financial support, and toMichael Atamas, Sarah Beckerman, and Noam Lerer for research assistance. The views expressedherein are those of the author and do not necessarily reflect the views of the National Bureau of EconomicResearch.

NBER working papers are circulated for discussion and comment purposes. They have not been peer-reviewed or been subject to the review by the NBER Board of Directors that accompanies officialNBER publications.

© 2012 by Louis Kaplow. All rights reserved. Short sections of text, not to exceed two paragraphs,may be quoted without explicit permission provided that full credit, including © notice, is given tothe source.

On the Optimal Burden of ProofLouis KaplowNBER Working Paper No. 17765January 2012JEL No. D81,K14,K41,K42

ABSTRACT

The burden of proof is a central feature of adjudication, and analogues exist in many other settings.It constitutes an important but largely unappreciated policy instrument that interacts with the levelof enforcement effort and magnitude of sanctions in controlling harmful activity. Models are examinedin which the prospect of sanctions affects not only harmful acts but also benign ones, on account ofthe prospect of mistaken application of sanctions. Accordingly, determination of the optimal strengthof the burden of proof, as well as optimal enforcement effort and sanctions, involves trading off deterrenceand the chilling of desirable behavior, the latter being absent in previous work. The character of theoptimum differs markedly from prior results and from conventional understandings of proof burdens,which can be understood as involving Bayesian posterior probabilities. Additionally, there are importantdivergences across models in which enforcement involves monitoring (posting officials to be on thelookout for harmful acts), investigation (inquiry triggered by the costless observation of particularharmful acts), and auditing (scrutiny of a random selection of acts). A number of extensions are analyzed,in one instance nullifying key results in prior work.

Louis KaplowHarvard UniversityHauser 322Cambridge, MA 02138and NBERmeskridge@law.harvard.edu

1. Introduction

All systems of adjudication must determine how high to set the burden of proof: thestrength of evidence required for the imposition of sanctions, award of damages, or provision ofother forms of relief. Proof burdens are set in other contexts as well: internal organizationaldecision-making (standards for promotion, employee discipline, or product launches),contractual requirements (defect rates for rejection of a supplier’s shipment), and medicaltreatment.

The familiar tradeoff of type 1 and type 2 errors, such as in medical treatment, focuses onthe costs and benefits of various post-decision outcomes. In many settings, however, includingfor the legal system, a (or the) primary purpose and effect of decisions and resulting actions isthe creation of ex ante incentives. The prospect of correct imposition of sanctions deters harmfulactivity; mistaken exoneration dilutes it. Moreover, the expectation that sanctions willsometimes be mistakenly applied to benign acts tends to chill desirable behavior. Consider, forexample, antitrust, where misapplication of sanctions may discourage efficient, pro-competitivebehavior (for example, promotional product pricing may look like predation); securitiesregulation, where the prospect of erroneous sanctions may increase the cost of capital (IPO’smay result in liability even in the absence of any misrepresentation); medical malpractice, whereworries about false positives may discourage cost-effective care (including denial of care tohigh-risk patients and reduced physician supply in certain fields); and contract breach generally,where concern for misassessment of performance may deter efficient contracting or misdirectbehavior governed by contracts.1 In these settings and others, choosing the burden of proofposes an error tradeoff, but a more complicated one than in standard problems like medicaldecision-making because the focus is on ex ante behavior, which is endogenous.

This article extends models of law enforcement by treating the proof burden as a policyinstrument alongside the traditional ones, enforcement effort and the level of sanctions. Becausethe results differ in important ways depending on the enforcement technology, three types ofmodels are analyzed. Monitoring – posting lookouts for harmful acts, such as police patrols orprivate security guards – is considered first because its features are elements of the other cases. Investigation – inquiry triggered by the costless observation of particular harmful acts, such aswith many crimes, accidents, and contract breaches – differs because benign acts can bemistakenly sanctioned only as a consequence of legal action triggered initially by thecommission of a harmful act. Auditing – scrutiny of a random selection of acts, such as taxexaminations, inspections for regulatory compliance, and intellectual property owners’ surveysof competitors’ products – is analytically identical to monitoring in terms of behavior but differsin administrative costs because greater activity (harmful and benign) requires more audits for agiven detection rate.

Among the more important findings applicable to all three models is that the

1For empirical evidence suggestive of the importance of these phenomena, see, for example, Walker et al.(2010) on IPO litigation risk; Studdert et al. (2005) on the avoidance of medical procedures and patients posing highlitigation risks; and Kessler, Sage, and Becker (2005) on medical malpractice and physician supply.

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determinants of the optimal evidence threshold depart qualitatively and substantially fromfamiliar understandings regarding the burden of proof – which involve consideration of thelikelihood that an individual before the tribunal committed a harmful act rather than a benignone. For example, it is demonstrated that raising the evidence threshold can increase thefrequency with which individuals who commit benign acts are subject to sanctions and thatraising this threshold can result in a lower ex post likelihood that an individual must havecommitted the harmful type of act in order to be subject to sanctions.

Section 2 presents the first model, involving monitoring, which is employed in themajority of the analysis because it is somewhat more streamlined and, as mentioned, nearlyevery effect also appears in the other models. There are two groups of individuals who maycommit acts that yield private benefits. For one group, the act causes external harm; for theother, the act is socially benign. The government chooses a level of enforcement that determinesthe fraction of acts that are scrutinized. As mentioned, the posting of monitors (police patrols)has this feature. At this first level, enforcement does not distinguish the two types of acts, adifference that is initially unobservable to the authority. But the process of scrutinizing actsultimately gives rise to evidence, which is then assessed to determine whether to imposesanctions. The evidence is modeled as a signal. The distribution of signals is higher for harmfulacts than for benign ones. The choice of a burden of proof can be thought of as selecting acutoff, where sanctions are applied if and only if the signal exceeds the chosen threshold. Any(nonextreme) rule entails both true and false positives and negatives.

Section 3 analyzes optimal policy, focusing on the level of enforcement effort and thesetting of the evidence threshold. As one would expect, the prospect of chilling desirablebehavior is central to both choices. Greater enforcement effort, in addition to involving the usualtradeoff of deterrence and enforcement costs, also raises the expected sanction on benign acts, sooptimal enforcement effort is lower on this account. The evidence threshold is set to equate themarginal effects of deterring harmful acts and chilling benign acts. The benefit of deterring aharmful act falls at the margin whereas the cost of chilling a benign act rises, althoughnonuniform distributions of benefits for each type of act can produce countervailing effects.

Additional attention is devoted to the respects in which greater enforcement effort and alower evidence threshold are substitute means of achieving deterrence. The former involvesdirect resource costs whereas the latter does not. Accordingly, for there to be an optimal mix, itmust be that a lower evidence threshold has some relative disadvantage. Under standardassumptions, a lower evidence threshold (compared to greater enforcement effort) causesrelatively more chilling per unit of deterrence enhancement – and, if it did not, one could not bein the neighborhood of an optimum. The analysis also suggests a new, complementary versionof the so-called Becker (1968) argument favoring maximal sanctions: it is generally optimal toraise sanctions (if they are not maximal) and simultaneously raise the evidence threshold,holding deterrence constant; the only effect of this combination is a reduction in chilling costs. Perhaps surprisingly, therefore, the prospect of mistakes makes it harder rather than easier torationalize the widespread practice of employing only moderate sanctions for less harmful acts. Moreover, this finding overturns key results in prior literature that abstracts from errors and,accordingly, does not feature the evidence threshold as an instrument.

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Section 3 also relates the present analysis, which focuses on the choice of an evidencethreshold, to conventional notions about the burden of proof, which may be understood in termsof Bayesian posterior probabilities. Consider, for example, the preponderance of the evidencestandard, often taken to mean that it must be more probable than not that the defendant beforethe tribunal committed a harmful act (which may include failing to take due care, breaching acontract, and so forth). The determinants of the optimal evidence threshold and of thepreponderance standard (or other ex post probability-based standards) are quite different. (Theydo have two common factors, but the other seven terms in the two expressions differ.) Comparative statics can also be counterintuitive. Raising the evidence threshold can result in alower ex post likelihood that an individual before the tribunal must have committed a harmful actin order to be subject to sanctions. Also, raising the evidence threshold can increase the absolutefrequency with which individuals who commit benign acts are mistakenly subject to sanctions.

The preceding points raise the question of how some prior work could have found thepreponderance of the evidence standard to be optimal. To match more closely those models, afinal subsection presents a modified, simpler version of the present model in which there are nobenign acts, so individuals only decide whether or not to commit a harmful act. In this variant,the main result in previous papers is immediate. Nevertheless, this result is not thepreponderance of the evidence standard but rather a statement about ex ante probabilitiesconditional on individuals’ behavior. Moreover, comparison of the single factor that determinesthe optimum in this modified model with the first-order condition for the optimal evidencethreshold in the central model here reveals that there is little relationship between the two (thelatter condition including six additional factors). Hence, intuitions gleaned from the sort ofmodel used in past work provide little guidance regarding the optimal evidence threshold in thesettings examined in this article.

Section 4 presents a second model, for the case in which enforcement is throughinvestigation rather than monitoring. Specifically, it is supposed that investigations are triggeredby the observation of harmful acts, the sighting of which does not identify the perpetrator. Forexample, for murder, auto theft, or groundwater contamination, it is often clear that a violationoccurred, and the authorities engage in investigation to determine who committed the harmfulact. In this setting, the nature of the optimum for both enforcement effort and the evidencethreshold depends on additional factors that were absent in the monitoring model. First,enforcement effort only needs to be expended when harmful acts occur, so deterrence reducesenforcement costs; this factor favors greater enforcement effort and a lower evidence threshold(the latter, by raising deterrence, similarly reduces enforcement costs). As a consequence, thereneed not be underdeterrence at the optimum, unlike when enforcement is by monitoring. Second, because deterrence reduces the frequency of investigations, it also reduces the likelihoodthat benign acts are mistakenly subject to sanctions and thus mitigates chilling costs. As well,chilling costs are in a sense lower to begin with since scrutiny is only triggered when harmfulacts are committed. A number of these considerations are favorable to greater enforcementeffort and a lower evidence threshold. Relatedly, when enforcement is by investigation, thefactors determining the optimal evidence threshold differ in yet additional ways (by comparisonto the model of monitoring) from the preponderance of the evidence standard or otherqualitatively similar rules. Also, as in the model with monitoring, raising the evidence thresholdcan increase the absolute frequency with which individuals who commit benign acts are

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mistakenly subject to sanctions, but the circumstances in which this is so differ qualitatively.

Section 5 offers four extensions. Auditing (a third model) is treated briefly because itsdifferences from the monitoring model are modest. Behavioral effects are the same, so manyresults from the monitoring model are applicable. However, enforcement costs are qualitativelydifferent, which increases the divergence between the determination of the optimal evidencethreshold and that which satisfies conventional burden of proof notions.

Costly sanctions are considered next. Their introduction might seem to favor aheightened evidence threshold because this reduces the frequency with which sanctions areimposed on both the harmful and benign acts that are committed. There is, however, thecompeting effect that raising the evidence threshold reduces the deterrence of harmful acts andthe chilling of benign acts, both of which increase the pool of individuals potentially subject tosanctions. Accordingly, the common view that the existence of costly sanctions obviouslyjustifies a tougher proof requirement in the criminal context is undermined. Incapacitationbenefits of sanctions (notably, imprisonment) as well as any social costs associated with the veryact of mistaken imposition of sanctions are each analyzed as variations in social sanction costs,with the implication that the effects of these considerations on the optimal evidence threshold aresimilarly ambiguous, again contrary to conventional wisdom.

The third extension considers the accuracy of evidence. First explored is how an increasein accuracy affects the optimal evidence threshold. It is explained that there is serious ambiguityin posing such a question and, moreover, that there are a number of competing influences, so noclear answer can be offered. Second, it is explained how one can use the present analysis toidentify the value of increased accuracy. Finally, a fourth extension addresses the relevance ofwhether there is a small, highly concentrated group of benign acts that might mistakenly enteradjudication versus a broad, more disbursed group, each less likely to be swept up in the legalsystem but with the same overall frequency of misidentification of benign acts. The formersituation tends to involve greater chilling costs and thus to favor a higher evidence threshold,although this conclusion need not hold due to nonconstancy of the density functions forindividuals’ benefits from acts.

Prior literature on the burden of proof largely considers different issues. One set ofarticles focuses on adjudication itself, such as regarding the resources devoted to the presentationof evidence or bringing suit (usually with exogenous underlying behavior).2 Some of this workhas emphasized the placement of the burden to go forward – that is, the specification of whichside loses if little or no evidence is presented – rather than the strength of the burden of proof.3 Another set of literature does consider underlying behavior but has largely been confined tomodels in which there is only one type of decision: whether or not to commit a harmful act (or,

2See Bernardo, Talley, and Welch (2000), Demougin and Fluet (2008), Hay and Spier (1997), Miceli(1990), Posner (1999), and Rubinfeld and Sappington (1987).

3In legal usage, the single phrase “burden of proof” is sometimes used to refer to the persuasion burden(how convinced the factfinder must be, such as under the preponderance rule) and also the production burden (whichparty has the burden of going forward, i.e., who loses if little or no evidence is presented), although recently theformer reference has become more common. See Garner (1999).

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similarly, whether or not to take due care).4 In such a model, the harm of mistaken imposition ofsanctions is simply to reduce deterrence by making the socially preferable option less attractive. Often, deterrence is taken to be inadequate, so the problem reduces to choosing the burden ofproof that maximizes deterrence. As already mentioned, these models (which are examined insection 3.E) are much simpler than the ones considered here, and their qualitatively differentresults provide little indication of optimal policy in the present settings.

2. Model I: Monitoring

There are two types of acts that may be committed, a harmful one, H, and a benign one,B. The harmful type of act imposes an external social cost of h; the benign type of act involvesno externality.5 A mass of individuals normalized to 1 may commit the harmful type of act. Those who may commit the benign type of act have a mass of γ; an interpretation is that γindicates the relative mass of benign acts that may be undertaken in situations in which theymight initially be confused with harmful acts, for other benign acts do not face the risk ofsanctions and thus are unaffected by the enforcement instruments under consideration. (Formany purposes, it would be possible to suppose that the same individuals may commit bothtypes of acts, and that γ indicates the relative frequency of opportunities to commit the benigntype of act.6) Individuals’ benefits from committing an act are b, with differentiable density andcumulative distribution functions f i(b) and F i(b), respectively, where i = H, B. Individuals knowwhat type of act they are able to commit and its benefits to them, but the government initially hasno knowledge of acts’ type and never learns individuals’ benefits from acts.

Throughout this article, the timing of the model is:1. The government sets enforcement effort, the sanction, and the evidence threshold.2. Individuals learn their type of act and their private benefit.3. Individuals decide whether to act.4. A portion of those who act are identified and brought before a tribunal.5. In adjudication, individuals are sanctioned if and only if the evidence in their case

exceeds the evidence threshold.

In this first version of the model, enforcement is by a government authority thatscrutinizes the fraction π(e) of acts, where e indicates enforcement effort (expenditures) and

4See Demougin and Fluet (2006, 2008), Kaplow and Shavell (1994), Lando (2002), and Polinsky andShavell (2007, pp. 427–29). Png (1986) has an additional behavioral dimension (activity level, in an accidentsetting) but does not analyze the burden of proof.

5It would be straightforward to adjust the analysis to allow it to involve a positive externality or a negativeexternality at a different level from that of the H type of act.

6The primary caveat pertains to setting the optimal sanction: specifically, the notion of setting the sanctionat its maximum may be problematic if individuals might be sanctioned for multiple acts. Another variation would beto allow individuals to choose one of the two acts or inaction. This variation would significantly complicate theexposition but have only a modest effect on the qualitative results. Note that, in this setting, deterrence of harmfulacts induces some individuals to switch to benign acts rather than inaction, and chilling of benign acts causes someindividuals to switch to harmful acts rather than inaction.

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πN(e) > 0.7 (Allowing type-specific π’s would not much affect the analysis, so is avoided forsimplicity.8) One might think of enforcement as involving the posting of monitors (such as withmuch traffic enforcement or other police patrols) who will identify as suspicious some fractionof the acts that are committed.

If an individual’s act is scrutinized, thereby leading to adjudication, the probability ofbeing subject to the sanction s – taken for now to be a monetary penalty that is socially costlessto impose – is pi. The government, unfortunately, is not free to choose pH and pB independently;if it could, it would set the former to 1 (or sufficiently high to achieve first-best deterrence ofharmful acts) and the latter to 0, thereby sanctioning all before the tribunal (or enough) whocommit harmful acts and none who commit benign acts. Rather, it is supposed that there is someset of evidence produced when an individual is scrutinized. The government can choose a lowerevidence threshold, which yields a higher pH, but only at the cost of a higher pB. Let us nowspecify this relationship.

The evidence in a given case will be represented by the variable x, a signal of theunderlying act.9 For each type of act, the densities and cumulative distribution functions for xare given by g i(x), assumed to be positive10 and differentiable, and G i(x), respectively, whereG B(x) $ G H(x), with strict inequality except when the G i equal 0 and 1 (a strong form of first-order stochastic dominance).

Suppose the government sets the evidence threshold at some xT, which is to say thatsanctions are applied if and only if the value of the signal x exceeds xT. Since G i(xT) denotes theprobability that the signal falls below the threshold, it follows that pi(xT) = 1!G i(xT). Ourassumption about the relationship between the two distribution functions implies that pH(xT) > pB(xT) – except when the G i equal 0 and 1. Regarding the choice of xT, note that it is equivalent(and sometimes useful for thought and to reduce notation somewhat) to suppose that thegovernment instead chooses pH. By inverting the function pH(xT), this implies a value of xT, andhence of pB(xT). Accordingly, we can also define an implicit function for pB that will be denoted

7We could further assume that πO(e) # 0, which would ordinarily be done to ensure that the second-ordercondition for welfare maximization with respect to e holds. As is apparent from the analysis in the appendix,however, it would still be necessary to confine attention to situations in which the second-order condition holdsbecause there are other terms (involving the derivatives of the density functions) that may have the wrong sign.

8If the π’s differed by a constant factor (for example, the hit rate on benign acts may be proportionally lowerthan that on harmful acts), then the only effect would be that this constant would appear analogously in each first-order condition and in the condition for the preponderance rule, without affecting any qualitative statements orresults. With general differences in the π(e) functions, allowed in section 4’s extension for enforcement byinvestigation, the only further effects are precisely those presented there: the Becker argument for maximal sanctionsneed not hold, as discussed following expression (12), and equation (6) on optimizing both enforcement effort andthe evidence threshold would become identical to equation (17).

9The optimal method of combining evidence (relevant in practice since multiple signals ordinarily arepresent) is given by likelihood ratios, as indicated by the Neyman-Pearson (1933) Lemma.

10They may equal zero at the lowest and highest possible values of x, if such exist (i.e., if the interval ofpossible values of x is closed at either end). The supports may be taken to be the same for signals associated withharmful and benign acts; for the problem to be interesting, all that is required is that the two distributions overlap.

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PB(pH).11 Because, from the definition of pi(xT), we have dpi(xT)/dxT = !gi(xT), note that we canalso write PBN = dPB/dpH = [dpB(xT)/dxT]/[dpH(xT)/dxT] = gB(xT)/gH(xT), which is positive. As willbe seen, the first-order condition for the optimal evidence threshold xT, (4) or (5), can be writtenusing only PBN, without explicit reference to the pi(xT), for all that matters is the rate at which pB

changes per unit change in pH. For some purposes, it is natural to assume further that PBO > 0,i.e., that the implicit function PB(pH) is convex. This feature, which is unnecessary for the resultsin this paper, is implied if the density gH(x) satisfies a strict version of the monotone likelihoodratio property with respect to gB(x).12

Finally, observe that choosing the evidence threshold xT seems akin to choosing a burdenof proof; a higher burden appears to correspond to a higher evidence threshold, and conversely. However, there is an important conceptual difference in that the threshold xT is defined in termsof the evidence (the value of the signal) whereas conventional notions of a burden of proof aredefined in terms of Bayesian posterior probabilities. Put another way, the pi(xT) refer to theprobabilities of being sanctioned conditional on having committed an act of type i and then beingscrutinized. These are ex ante probabilities that depend on one’s type of act. By contrast, indiscussions of the burden of proof, it is ordinarily understood that some act is before a tribunal,and we are inquiring into the probability that it is of the harmful type rather than the benign type. This ex post probability depends on the particular signal received (a specific value of x in theforegoing terminology, which is used to update one’s prior) and on the base rates of the activity(the relative levels constituting one’s prior), the latter of which are endogenous (notably, theydepend on how the proof burden itself is set). We shall return to this matter in section 3.D.13

Individuals are assumed to be risk neutral, so those whose type of act is i commit their actif and only if b > π(e)pi(xT)s, that is, their benefit exceeds the expected sanction for their type ofact.14 Social welfare, W, is taken to be the aggregate of individuals’ benefits from acting minus

11An equivalent expression is commonly used in signal detection theory, the graph of PB(pH) correspondingto what is called the receiver operating characteristic (ROC) curve. See Luce (1963).

12See Milgrom (1981). If PBO < 0 for some values of pH, then a term in the second-order conditions (see(A1) and (A6) in the appendix) is positive, but since other terms may also be positive, first-order conditions are onlynecessary conditions and, for the social optimum, attention must in any case be confined to cases in which thecorresponding second-order conditions are confirmed to hold. Part of why the assumption that PBO > 0 is natural toimpose is that, otherwise, the optimal evidence rule does not consist of a single threshold xT because one shouldapply sanctions for signal values with the highest likelihood ratios first (as stated by the Neyman-Pearson (1933)Lemma. Note that, when the assumption fails, one could reproduce the optimality of a single cutoff rule byreordering the values of the signal in accordance with the likelihood ratio, in which case the monotone propertywould hold for the transformed signal. On the relationship between the monotone likelihood ratio property and theoptimality of simple cutoff decision rules, see Karlin and Rubin (1956).

13To foreshadow the analysis, any evidence threshold will in equilibrium generate a resulting,conventionally defined, burden of proof; that is, at the evidence threshold xT, there is some corresponding ex postlikelihood that the act is of the harmful type. The converse (moving from a stipulated ex post likelihood to anevidence threshold) is trickier – inversion would be possible if the relationship between the two was monotonic,continuous, and spanned the relevant range. The key complication is that the application of conventionallyunderstood proof burdens depends on behavior, which itself depends on the proof burden.

14Introducing risk aversion would have two main effects. First, raising s would increase the deterrence andchilling punches of sanctions at an increasing rate. Second, sanctions would then be socially costly, as considered inthe extension in section 5.B (although with risk aversion the social cost per unit of the sanction would be nonlinear

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the harm from the commission of acts of type H and the cost of expenditures on enforcement.

( ) ( ) ( ) ( ) .( ) ( ) ( ) ( )

1 W b h f b db bf b db eH

e p x s

B

e p x sH T B T

The first term indicates the benefits and harm attributable to harmful acts, and the second thebenefits from benign acts. In each case, the lower limit of integration is the benefit of theindividual just at the margin, with all individuals having greater benefits committing the act.

3. Analysis of Model I

The government’s problem is to choose e, xT, and s in a manner that maximizes socialwelfare (1). It is immediate that the optimal sanction is maximal. Following the idea in Becker(1968), if the sanction is not maximal, one could raise s and lower e by an amount that keeps theproduct π(e)s constant, in which case there would be no effect on either of the first two terms inexpression (1), but welfare would rise on account of e being lower. For most of the analysis, thesanction s will simply be taken as given. It might be supposed to be at its maximal level, or,alternatively, it may be fixed for other reasons.

A. Optimal Enforcement Effort

To determine optimal enforcement effort, we can differentiate social welfare (1) withrespect to e. Because π is a function of e, enforcement’s effect on behavior is reflected in raisingthe lower limits of integration, and this is the source of deterrence benefits and chilling costs. The first-order condition (suppressing some arguments of functions) may be expressed as

( ) .2 1 p sf p s h p s p sf p s p sH H H H B B B B

It is convenient to define bi / πpis, the benefits of the individuals just indifferent about whetherto act, in which case expression (2) can be restated as

( ) .3 1 p sf b h b p sf b bH H H H B B B B

The left side represents the marginal gain from raising enforcement effort, which is the productof the deterrence punch πNpHs f H(bH) (the rise in expected sanction times the density ofindividuals just deterred per unit increase in expected sanction) and the net gain from deterringthe marginal harmful act: avoiding the social harm h but losing the private benefit bH from theact. The right side indicates the marginal cost from raising enforcement effort. The first term isthe cost of chilling desirable behavior, the product of the relative mass γ of the benign act in the

in the sanction). For a discussion of prior literature on how risk aversion influences optimal sanctions, see Polinskyand Shavell (2007, pp. 413-16).

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population times the chilling punch πNpBs f B(bB) (the rise in expected sanction on the benign acttimes the density) times the benefit bB from the marginal benign act that is forgone due to thehigher (mistaken) expected sanction. The final term is just the cost of the added unit ofenforcement effort.

Proposition 1: Given xT, at an interior optimum for the level of enforcement, e*:a. Raising h raises e*;b. Raising γ reduces e*;c. Raising s has an ambiguous effect on e*; andd. At e*, h > bH (i.e., there is underdeterrence relative to the first-best), and this would

be so even if enforcement were costless.

Propositions 1.a and 1.b are intuitive but the demonstrations are not entirelystraightforward. Raising h naturally favors higher effort because deterrence of harm is moreimportant. Raising γ favors lower effort because chilling effects are relatively more significant. Some complexity arises, however, because the pertinent derivatives of the first-order conditioninclude a number of terms of differing and in some cases indeterminate signs; relatedly, multipleinterior optima for e are possible. Much of the complication is due to the possibility that thedistributions of benefits are not uniform, which also implies that the second-order condition(expression (A1) in the appendix) need not hold globally. Suppose, for example, that bothdensities are single-peaked, with the benefits of benign acts concentrated at a low level and thoseof harmful acts at a higher level. Between the peaks, f B would be falling and f H rising, theformer implying that the chilling punch of greater enforcement is falling and the latter that thedeterrence punch of greater enforcement is rising. If these effects from the density functions aresufficiently large relative to the others, one could have a local minimum in such a region. Nevertheless, as demonstrated in the appendix, the intuitive results obtain because, given thateffort is at an optimum, the second-order condition implicitly limits the magnitude of effects dueto the nonconstancy of the density functions.15

Regarding proposition 1.c, raising s has multiple influences. First, it raises the bi, both ofwhich favor lower enforcement effort. For harmful acts, the marginal deterrence gain is fallingwith the expected sanction (and thus with e or s considered separately, holding everything elseconstant) because the harm h is the same across harmful acts but the forgone benefit fromdeterring the marginal individual who otherwise commits the harmful type of act, bH, is rising. For benign acts, the marginal chilling cost is rising with the expected sanction because theforgone benefit from chilling the marginal individual who otherwise commits the benign type ofact, bB, is rising. Second, raising s boosts the productivity of greater enforcement effort, whichfavors higher enforcement. This second effect, by itself, is not ambiguous at the optimum:although both marginal deterrence and marginal chilling rise, the welfare effect of the formernecessarily dominates that of the latter because the former must equal the sum of the latter andthe “plus one” that consists of the effort cost itself. Third, as was the case with propositions 1.aand 1.b, nonconstant density functions create additional effects, each of which in general can

15Note also that it is possible for there to be a corner solution involving e* = 0 if the marginal return fromenforcement, πN(0), is sufficiently low relative to the magnitude of h.

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have any sign. In the appendix, examples are constructed in which raising s raises e* and inwhich it reduces e*.

Proposition 1.d follows from inspection of expression (3). Because the right side isnecessarily positive, so must be the left side at the optimum, which indicates that h > bH, whichis to say that the harmful type of act is underdeterred relative to first-best behavior. This result iscommon in models of optimal enforcement due to the positive cost of raising enforcement effort. Here, there is the additional reason, the chilling cost (the first term on the right side), whichimplies that the optimal degree of underdeterrence is greater. Even if additional enforcementwas free (in which case there would be no “plus one” on the right side of 3), one would stopshort of first-best deterrence.

B. Optimal Evidence Threshold

Our main policy variable of interest is the evidence threshold xT. If we take thederivative of social welfare (1) with respect to xT – keeping in mind that the pi are functions of xT

so that the evidence threshold’s effect is through reducing the lower limits of integration – weobtain a first-order condition that can be expressed as

( ) ,4 sf p s h p s sP f p s p s orH H H B B B B

( ) .5 f b h b P f b bH H H B B B B

Lowering xT is costless in terms of direct resource costs; its effects on social welfare areexclusively through behavior. On the left side of (4) or (5) is the social gain: the deterrencepunch, indicated by the height of the density for those just deterred, times the net gain fromdeterring the marginal harmful act. (Note that πs was divided out of both sides in moving from(4) to (5), for convenience of later comparisons.) The right side indicates the social cost: thechilling punch – the relative mass of population that contemplates committing the benign acttimes the rate that pB rises per unit of increase in pH (as a consequence of the reduction in xT),times the density – all multiplied by the net benefit forgone from chilling the marginal benignact.

Proposition 2: Given e, at an interior optimum for the evidence threshold, xT*:a. Raising h reduces xT*;b. Raising γ raises xT*; andc. If the distributions F i are uniform, raising s raises xT*.

These proofs are also in the appendix, but the core basic reasoning is articulated here. Raising h naturally favors a lower level of xT because deterrence is more important. Raising γfavors a higher xT because chilling is more significant. And, as with proposition 1, theseintuitive results hold despite complications regarding nonconstancy of the density functions.

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For proposition 2.c, note first that if πs is higher – whether due to higher enforcementeffort or sanctions – then the optimal evidence threshold xT will tend to be higher because, asexplained with regard to proposition 1.c, the marginal net benefit of deterring harmful acts isreduced when deterrence is already higher and the marginal cost of chilling benign acts is raisedwhen the marginal chilled individual’s benefit is higher. This claim is proved in the appendixwith regard to changes in s for constant density functions. Nonuniform distributions of thebenefits from either of the two types of acts complicate this phenomenon (but not so much 2.aand 2.b because, again, satisfaction of the second-order condition, expression (A6) in theappendix, guarantees the claimed properties). A higher f H (locally) will favor a lower xT, and ahigher f B (locally) will favor a higher xT. The analysis of how changing the sanction influencesthe problem therefore depends on how the densities are changing locally. Specifically, if f H isrising rapidly, then a slight increase in s notably increases the marginal deterrence punch,favoring a lower xT; likewise, if f B is falling rapidly, then a slight increase in s notably reducesthe marginal chilling punch, favoring a lower xT. Accordingly, statement of the unambiguousresult in proposition 2.c is limited to the case of a uniform density function.

C. Comparison of Raising Enforcement Effort and Lowering the Evidence Threshold

It is illuminating to compare directly the choice between raising enforcement effort e andlowering the evidence threshold xT as ways to deter harmful acts. Laxer proof requirements andgreater enforcement effort are substitutes, but imperfect ones that differ along two dimensions. Compare the first-order conditions (5) and (3). First, as previously noted, reducing xT, whichraises pH, is free of resource costs in the present setting, unlike enforcement effort. Second,abstracting from the densities, which are analogous in the two conditions, the behavioral punchesare in the proportion pH to pB in (3) for enforcement effort but they are in the proportion 1 to PBNin (5) for the evidence threshold.

To represent these ideas explicitly, use the first-order condition (3) for optimalenforcement effort e to substitute into condition (5) for the optimal evidence threshold xT. Oneway of doing so yields

( ) .61

P

p

pf b b

p sB

B

HB B B

H

The term in parentheses on the left side depicts the difference in the relative “effective” rise in pB

as one “effectively” increases pH: that for raising pH directly as a consequence of reducing xT

minus that for raising enforcement effort, which does not raise pH or pB per se but does increasetheir common multiplier π. This term is then multiplied by the marginal importance of chillingeffects (the leading γ indicating relative importance of the benign type of act, also multiplied bythe density and the forgone benefit from the marginal benign act that is chilled). In other words,the left side is the relative difference in chilling effects weighted by their importance. (Note that,in this formulation, the deterrence of harmful acts is implicitly held constant.) On the right sideis the cost disadvantage of raising enforcement effort, normalized in units of deterrence punch(rise in the expected sanction on harmful acts). When both enforcement effort and the evidence

- 11 -

threshold are optimized, these magnitudes are equated.

Proposition 3: Higher enforcement effort and a lower evidence threshold are substitutes inachieving a given level of deterrence, and the optimal mix of enforcement effort and evidencethreshold is given by expression (6).

Note further that the right side of (6) is necessarily positive, so it must be that, at theoptimum, PBN > pB/pH.16 This feature indicates that, for a given increase in πpH, there is arelatively greater rise in πpB when one raises pH directly, by lowering the evidence threshold xT,than when one raises its ultimate impact by raising π, that is, by increasing enforcement effort. In this respect, lowering the evidence threshold xT is a less advantageous means of increasing thedeterrence of harmful acts than is raising enforcement effort e. The intuition is that lowering theevidence threshold assigns liability in cases of lower quality (a lower ratio of ones with harmfulacts to ones with benign acts) than those previously giving rise to sanctions (which had highervalues of x). By contrast, raising enforcement effort brings more cases into the system across theevidence-quality spectrum, producing liability in cases of the same average quality as before.

This result can also be related to the comment at the outset of section 3 that optimalsanctions are maximal in this model. Raising sanctions dominates lowering the evidencethreshold because it has the same superior discriminating power as does raising enforcementeffort (but it is costless). Specifically, for a given increase in s, if xT is increased to keep pHsconstant, the requirement that PBN > pB/pH implies that pBs falls. This point can be seen as a newversion of the argument associated with Becker (1968) that favors maximal sanctions –traditionally, because it is generally optimal to raise sanctions and lower enforcement effort inachieving a given level of deterrence.17 Here, the savings involve a reduction in chilling effectsrather than in enforcement expenditures. It is interesting that the possibility of mistakenimposition of sanctions provides an additional reason favoring high sanctions even for actscausing low harm, which adds to the already serious tension between this prescription of manymodels of optimal law enforcement and observed practice.

Indeed, this consideration nullifies two of the main results in important papers byMookherjee and Png (1992) and Shavell (1991). In their models, the same monitors observemultiple kinds of harmful acts whose degrees of harm differ (perhaps littering and mugging). When the single level of enforcement effort is optimized, the less harmful types of acts might befully deterred by a less-than-maximal sanction, in which event the optimal sanction for such actsis set to achieve first-best deterrence. However, if one introduces benign acts that might bechilled and takes the evidence threshold to be an enforcement instrument, it instead would beoptimal to raise sanctions to the maximum even for low-harm acts, while simultaneously raisingthe evidence threshold for them, in order to reduce chilling costs, as just explained. Chilling

16If one assumes that PBO > 0, this condition is necessarily satisfied (even if not at the optimum, as long asthe probabilities are not 0).

17And, just as the traditional Becker argument is known to imply that, as the maximum feasible sanctiongoes to infinity, optimal enforcement effort approaches zero, we have here that one can indefinitely raise theevidence threshold, driving down chilling effects.

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costs also make underdeterrence optimal even though first-best deterrence can be achievedcostlessly (recall proposition 1.d).

D. Comparison to Conventional Burden of Proof Notions

The foregoing results should be compared to those under, say, a preponderance of theevidence standard that is used in civil adjudication in the United States (and which has been thesubject of some previous literature using quite different models, examined in section E).18 Undera typical burden of proof notion, one asks whether, given the evidence, the likelihood that theindividual before the tribunal committed the harmful act (the Bayesian posterior probability)exceeds some level, 50% under the preponderance standard. Suppose, then, that the tribunalobserves signal x. Conditional on the individual’s act being of type i (and the individual beingscrutinized, where individuals committing both types of acts are scrutinized with the sameprobability π), the likelihood of the signal is given by the density gi(x). The probability that agiven individual before the tribunal committed the harmful rather than the benign type of act alsodepends on the base rates of the activities, 1!F i(bi), further multiplied by γ for the benign act.The bi’s, of course, are not parameters but are determined endogenously by individuals’decisions about acts, which in turn depend on the enforcement parameters, including theevidence threshold (burden of proof) itself.

To identify the proper threshold for the signal x if one wishes to implement apreponderance of the evidence rule, we can ask what value of xT implicitly solves the equation:

( ) ,7 1 1g x F b g x F bH T H H B T B B

where the bi’s are determined assuming the use of this xT (and some given π and s), with theimplied values of the pi(xT). In expression (7), the left side indicates the frequency with whichindividuals who committed harmful acts (conditional upon being scrutinized and thus appearingbefore the tribunal) will be associated with the signal xT, and likewise for the right side forindividuals who committed benign acts. If the frequencies represented by the two sides ofequation (7) are equal, then indeed xT is a value of the signal such that it is equally likely that theindividual before the tribunal committed each type of act. Accordingly, insisting that x exceed xT

satisfies the more-likely-than-not (preponderance) standard (with additional assumptions on thegi).19 Note that if we desired a different likelihood at the threshold xT, we could multiply by afurther constant to the right side. For example, for a requisite likelihood of 75%, implying thatthe threshold signal was three times more likely to have been emitted from an individual whocommitted the harmful type of act than one who committed the benign type of act, one wouldmultiply the right side by 3. Or if proof beyond a reasonable doubt were understood to require,say, a likelihood of at least 90% or 95%, one would multiply the right side by 9 or 19,

18For extensive discussion of how the present analysis applies to legal discussions of the burden of proof,see Kaplow (2012a).

19A sufficient condition is that PBO > 0, which, recall from section 2, is implied by the density gH(x)satisfying a strict version of the monotone likelihood ratio property with respect to gB(x). See note 12.

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respectively.

To facilitate comparison with the equality determining the optimal evidence threshold, itis helpful to restate expression (7). Recall from section 2 that PBN = dPB/dpH =[dpB(xT)/dxT]/[dpH(xT)/dxT] = gB(xT)/gH(xT). Using this fact and rearranging terms yields

( ) .8 1 1

F b P F bH H B B B

Now compare expression (5), reproduced here for ease of comparison:

( ) .5 f b h b P f b bH H H B B B B

Both have the constant γ on the right side, and both have the term PBN. In all other respects,however, these expressions differ markedly. The densities on each side of (5) are for thedistribution of benefits evaluated for the marginal individual, but no corresponding terms appearin (8). By contrast, each side of (8) is weighted by the base rate of the activity, indicated by oneminus the value of the cumulative distribution function; no such terms appear in (5). Thisparticular difference reflects the Bayesian, information-inference character of the conventionalburden of proof notion, in contrast to the optimizing, behavioral orientation of the first-ordercondition, where it is the density functions for benefits that matter because they determine themarginal behavioral effects. Additionally, the level of harm and the benefits of the two types ofmarginal individuals appear directly in (5) but not in (8). In short, setting an optimal evidencethreshold xT and finding the cutoff that corresponds to any given conventional burden of proofnotion are very different enterprises.

Proposition 4:a. The optimal evidence threshold, given implicitly by (5), generally differs from the

preponderance of the evidence rule, given implicitly by (8) (or any similar rule denominated interms of an ex post likelihood that an individual before the tribunal must have committed theharmful type of act).

b. At the optimal evidence threshold, the minimum ex post likelihood that an individualmust have committed the harmful type of act in order to be subject to sanctions can take on any(admissible) value.

c. Raising the evidence threshold xT may increase the frequency with which individualswho commit benign acts are mistakenly subject to sanctions.

d. Raising the evidence threshold xT may result in a lower ex post likelihood that anindividual must have committed the harmful type of act in order to be subject to sanctions.

Proposition 4.a having been established, turn to 4.b, stating that when the evidencethreshold is set optimally, the minimum ex post likelihood that an individual must havecommitted the harmful type of act in order to be subject to sanctions can take on any admissiblevalue (i.e., between zero and one). To confirm this conclusion, note that condition (5) for theoptimal burden of proof does not depend on the cumulative distribution functions F i whereas

- 14 -

condition (8) depends on these functions and on little else. Hence, for a given solution tocondition (5), one could make inframarginal adjustments to the distributions of benefits for thetwo types of acts to produce different values for the minimum ex post likelihood.20 Perhapsmore intuitively, observe that condition (5) implies that, as h goes to zero, the optimal evidencethreshold will approach infinity (when all acts produce positive net social benefits, or at worstnegligible net social harm, it should be essentially impossible for evidence to be strong enoughto apply sanctions). Likewise, as h approaches infinity, the optimal evidence threshold willapproach negative infinity (or the lower bound of the support).21

Proposition 4.c states that raising the evidence threshold xT can increase on the frequencyof mistaken applications of sanctions. Following traditional thinking about proof burdens, onemight suppose that they would fall, but this need not be so. On one hand, raising xT does make itless likely that unchilled individuals will be sanctioned for the benign acts that they commit. Onthe other hand, raising xT reduces the chilling effect, so more individuals commit benign acts andthus may be subject to sanctions. Specifically, it can be shown that, if and only ifbBf B(bB) > 1!F B(bB), the latter effect is greater and thus a higher evidence threshold raises thenumber of false positives.

Before demonstrating proposition 4.d, consider first whether, for any stipulated requisiteex post likelihood, there exists an xT that yields that likelihood. As one varies the value of xT, arange of ex post likelihoods will be traced; since the model’s assumptions guarantee that allbehavior changes continuously, any ex post likelihood will be possible, subject to the minimumand maximum values that result from this exercise. The full range of ex post likelihoods,however, cannot necessarily be produced for a given set of parameters. Most obviously, for arequisite ex post likelihood of 100%, there would have to be no benign acts committed. Notefurther that the relationship between xT and the requisite ex post likelihood need not bemonotonic (given our lack of tight restrictions on the shape of the f i), which implies that, for agiven likelihood, there may exist more than one choice of xT that would generate it. To illustratethis point, suppose that the distribution of benign acts was concentrated at a very low level, butthat for harmful acts was not. As xT is reduced, which increases pB, eventually most benign actswill be chilled, so the ex post likelihood will approach 100%. From there, demanding strongerevidence, that is, raising the threshold xT, would in such a case (for at least some range) reducethe ex post likelihood. This establishes proposition 4.d.

E. Modification: Model with No Benign Acts

As mentioned in the introduction, prior literature (cited in note 4) offers a prescription forthe optimal evidence threshold that differs markedly from that derived here. Moreover, some ofthis work purports to find that the preponderance rule is optimal. To explain these differences,consider a simpler model in which there are no benign acts – only harmful acts and a failure to

20One way to do this is to consider a case in which the two distributions are uniform; the endpoints of eachof the intervals can then be shifted (while keeping the pertinent bi’s in the intervals) to produce any desired result.

21As explained just below with regard to proposition 4.d, spanning the full range of xT need not be sufficientin a given case to guarantee that all admissible ex post likelihoods are spanned.

- 15 -

act. Suppose further, as in these other papers, that scrutiny covers inaction. Accordingly, theprobability of mistaken sanctions, pB, is now taken to apply to inaction (but for convenience thesame notation is retained).22

Specifically, all individuals are scrutinized with probability π(e). Those who commit theharmful act, as before, are (conditional on being scrutinized) subject to the sanction s withprobability pH whereas those who do not commit the act (but are scrutinized) now face thesanction with probability pB. Accordingly, an individual will commit the act if and only ifb!πpHs > !πpBs, which is equivalent to the condition b > π(pH!pB)s. Because individuals whodo not commit the harmful act might be sanctioned in any event, deterrence is reduced. (There isno other cost to mistaken sanctions; the only benign behavior that is chilled is inaction.) It isconvenient to represent pH!pB by Δp, since only the difference in the two probabilities matters.

Social welfare in this modified problem, denoted by a “~”, is

( )~

( ) .( ) ( )

9 W b h f b db eH

e p x sT

The optimal xT is determined implicitly by

( ) .10 1 0s P f b h bB H H H

Obviously, the optimum is characterized by PBN = 1, which is to say that Δp, the gap between thepi, is maximized (assuming that π and s are such that there is underdeterrence, which is optimalsince raising π is costly; recall proposition 1.d). As mentioned in the introduction, this result isessentially equivalent to the core finding on optimal proof burdens in prior models that in arelevant sense have only one act.23

22See also Kaplow (2011).23In some, the choice variable is a dichotomous level of care – engagement in the pertinent activity being

taken as given – where it is always desirable to take greater care and underdeterrence is present, making it optimal tomaximize the difference between the likelihood of the sanction when taking low care versus high care. A relatedfeature of these sorts of models is that the costs of false positives and of false negatives are the same: reducing pH

hurts deterrence by the same degree as does raising pB by the same amount. Thus, the practice in some work ofarbitrarily assigning the same costs to each type of error is unproblematic in such a model, although in the modelexplored in the rest of this article, the costs of the errors are not symmetric. Moreover, it is sometimes remarked thatit is problematic to assign a given cost to a type of error since the social cost of the error is nonconstant. That is trueeven in the model in this subsection: the cost of either type of error is reduced deterrence, which has a net social costof h!bH for the marginal act. As one changes the evidence threshold (or other control variables), this net marginalcost is changing. But that change is also irrelevant because, throughout, it remains symmetric (so stipulating equallosses for each type of error yields the correct answer for the optimum even if the assigned levels of the losses areincorrect – for setting the burden of proof, but not for setting, say, the optimal level of enforcement effort). Bycontrast, in the main model used in the rest of section 3, the error costs are not merely nonconstant but most typicallychange in opposite directions: as one raises enforcement effort or reduces the evidence threshold, the benefit of

- 16 -

We can now compare this result with the first-order condition (5). Although (5) has PBNon the right side, it also has six other components. Therefore, the optimal evidence threshold inprior literature indeed differs radically from that derived in this article’s model, as a consequenceof introducing the possibility of chilling benign acts.

Finally, consider implications regarding the preponderance of the evidence rule. Theanalogue to expression (8) for the present case is

( ) .11 1

F b P F bH H B H H

Some literature has stated that the preponderance rule implies the maximization of Δp, which wejust saw entails choosing xT such that PBN = 1. However, this result is not implied by expression(11), which instead equates PBN with the ratio of the base rates. For example, for acts that fewpeople commit, this ratio would be near zero, implying an extremely high evidence threshold(assuming that PBO > 0), that is, PBN much less than 1. What is really contemplated, if deterrenceis to be maximized, is that base rates be ignored. Indeed, sometimes reference is made toapplication of a modified preponderance rule under which equal base rates (or flat or unbiasedpriors) are stipulated.24 In that case, expression (11) would be equivalent to PBN = 1. Nevertheless, the preponderance rule is an ex post Bayesian concept, one that fundamentallyconflicts with ignoring ex post likelihoods and considering instead deterrence maximization,which here depends only on ex ante probabilities, the pi (to be precise, the ratio of theirderivatives). This semantic point is reinforced by considering the main model and the analysisof the preponderance rule in section D. If one examines expression (8) and assumes equal baserates, one again produces the conclusion that PBN = 1, which, as mentioned, bears littleresemblance to the optimal rule given by expression (5).

4. Model II: Investigation

A. Model

The model presented in section 2 supposes that the method of enforcement is akin to theposting of monitors that enable the authority to scrutinize the fraction π(e) of all acts (of bothtypes, since initially it cannot tell the difference). Now suppose instead that enforcementinvolves undertaking investigation when a harmful act is committed, the assumption being thatwhether a harmful act occurred can be readily observed (for example, murder, automobile theft,or ground water contamination) but not who committed the harmful act. Let πH(e) denote thefraction of harmful acts in which the correct individual is scrutinized, i.e., the person whoactually committed the harmful act. Further suppose that, for each harmful act, the likelihoodthat an individual who committed the benign type of act is subject to scrutiny is

deterring an additional harmful act is falling whereas the cost of chilling an additional benign act is rising (with thetotal marginal social consequence also depending on the magnitude of the pertinent density function, each of whichmay be rising or falling).

24See, for example, Demougin and Fluet (2006, pp. 970–71).

- 17 -

πB(e)γ(1!FB(bB)). The rationale for the additional factors, elaborated further just below, is thatan investigation is more likely to lead to the apprehension of an individual who committed abenign act when there are more such individuals. It is further assumed that πiN > 0, πiO < 0 fori = H, B.25 As before, conditional on scrutiny, the probability of being subject to the sanction s ispH if the individual before the tribunal actually committed the harmful act and pB if the individualbefore the tribunal committed a benign act, with the magnitude of each determined by theevidence threshold xT (and so forth). Finally, the timing is also as in model I: the governmentsets e, s, and xT; next, individuals learn their type of act and their private benefits, after whichthey decide whether to act; then, a portion are scrutinized and, in adjudication, are sanctioned ifand only if x > xT.

Once again, the expected sanction for those who commit the harmful type of act is πHpHs,which likewise indicates the benefit level bH of the marginal individual. For the benign type ofact, the analysis is more involved. Investigations are only potentially triggered with probability1!FH(bH). (This is also true for those committing harmful acts, but when an individual commitsa harmful act, that person knows that an investigation may be triggered, leading to scrutiny of theactor with probability πH(e).) Conditional upon an investigation, the probability that someindividual who committed a benign act is scrutinized is, as mentioned, πB(e)γ(1!FB(bB)). Accordingly, the ex ante frequency with which individuals who commit benign acts are subjectto sanctions is (1!FH(bH))πB(e)γ(1!FB(bB))pB. Now, since γ(1!FB(bB)) individuals commitbenign acts, the ex ante probability that any given individual who commits a benign act issanctioned is (1!FH(bH))πB(e)γ(1!FB(bB))pB /γ(1!FB(bB)) = (1!FH(bH))πB(e)pB, which willsometimes be referred to as ρB.26 Thus, the expected sanction for those who commit the benignact is (1!FH(bH))πB(e)pBs, or ρBs, either of which likewise indicates the benefit level bB of themarginal individual.

At this point, we can state the social welfare function for this modified problem.

25It may also be natural to suppose that πH + γπB < 1.26The assumption that the likelihood an individual committing the benign act is scrutinized is

πB(e)γ(1!FB(bB)) is chosen in part for convenience, which is now apparent. It does seem natural to allow thisprobability to be rising (even if not linearly) in γ(1!FB(bB)). A more general approach, which also incorporates thelater need to divide the expression by γ(1!FB(bB)), would employ the function πB(e,γ(1!FB(bB))). If this was doneinstead, the only effect on the analysis would be that expressions (14) and (16) for the derivatives of ρB with respectto e and pH, respectively, would each have an additional term (the same term) dividing them, indicating that themagnitude of the derivative would be dampened (enlarged) if the derivative of this modified πB with respect to itssecond argument was positive (negative), basically allowing for a feedback effect of marginal changes in the chillingeffect on the ex ante likelihood that individuals committing the benign act are scrutinized. Accordingly, little ofanalytical consequence is sacrificed by employing the particular functional form adopted in the text.

- 18 -

( ) ( )

( ) ( ) ( ) .

( ) ( )

( ) ( ) ( ) ( )

12

11

W b h f b db

bf b db F e p x s e

H

e p x s

B

F e p x s e p x s

H H H T

H H T

H H H T B B T

This expression differs in two ways from expression (1) for social welfare in the original(monitoring) version of the model. First, the lower limit of integration for the second termreflects the immediately preceding discussion concerning the likelihood that individuals whocommit the benign act will be subject to sanctions; in particular, this probability includes thefactor 1!FH(bH) because investigations are only triggered when harmful acts are committed. Second, the enforcement effort cost e has a similar weighting.

Before proceeding, note that the standard argument associated with Becker (1968) thatdemonstrates the necessary optimality of maximal sanctions need not hold in this extension. Asone raises s and lowers e so as to keep the deterrence of harmful acts constant, it is notnecessarily true that the chilling of benign acts is also kept constant (and it may rise) because itis not assumed that πHN/πH = πBN/πB. The new variant of the argument for maximal sanctionsintroduced in section 3.C likewise does not suffice to guarantee the optimality of maximalsanctions for similar reasons, unless further assumptions are introduced.27 In any case, for theremainder of this section, the sanction s will be taken to be fixed. The government’s problemwill be to choose e and xT to maximize social welfare (12).

B. Optimal Enforcement Effort

The first-order condition for enforcement effort e may be expressed (by analogy toexpression (3) for the original model) as

( ) ,13 1 H H H H HB

B B B H Hp sf b h b ed

desf b b F b

where

27If one raises a nonmaximal sanction s and raises the evidence threshold xT so as to keep deterrenceconstant (specifically, pHs is kept constant, and πH is unaffected), the value of the first integral and of the third termdo not change; the resulting reduction in pB, however, may not be great enough to offset the effect of the increase ins on pBs, so chilling can rise. In model I, condition (6) sufficed to guarantee that PBN > pB/pH at the optimum and thusthat pBs would fall. However, the analogue to expression (6) for the investigation model, expression (17) below,differs. As stated in the text, the traditional Becker argument can fail when πHN/πH > πBN/πB. In that situation, πB/πH

> πBN/πBN, so the term in large parentheses in (17) being positive does not guarantee that PBN > pB/pH.

- 19 -

( ) .14 1d

dep F b p p sf b

BB B H H B B H H H H

These expressions differ from expression (3) in a number of ways. On the left side of expression(13), there is the additional “plus e” term in brackets, which indicates that there is a greatermarginal benefit from deterring harmful acts, namely, that fewer investigations are undertaken,which saves enforcement expenditures. Second, on the right side, instead of a “plus 1” term, wehave 1!FH(bH), indicating that the additional enforcement effort is weighted by the fraction ofundeterred harmful acts, since only those trigger investigations.

Inspection of expression (14) reveals two additional differences. First, the effect ofincreased effort on chilling, reflected in the term leading with πBN, is likewise weighted by1!FH(bH), reflecting that chilling is muted since only acts causing harm trigger investigations(that might in turn go awry by ultimately resulting in sanctions on benign acts). Moreover, thechilling effect that does exist is reduced, as reflected in the second term, on account of greaterdeterrence of harmful acts. That is, increased effort, by deterring harmful acts, reducesinvestigations at the margin, which further reduces chilling costs. (Because of its genesis, it isnatural to think of the second term in expression (14) as associated with the left side of (13)rather than the right side, a point reinforced by comparing the terms explicitly.)

In light of these differences, it seems natural to view enforcement effort as more valuableat the margin when enforcement is by investigation. However, the two models are, at heart, notcomparable, because they reflect different technologies applicable in different circumstances. For example, nothing has been said about the relationship between π(e) in our original model andthe πi(e)’s in the present model. Nevertheless, these qualitative differences do suggest that thedifferent enforcement methods have different relative strengths and weaknesses, many of whichpertain to chilling effects in particular.

Finally, observe that, unlike in our original setup and in many standard law enforcementmodels, it need not be true that there is underdeterrence (relative to first-best behavior) at theoptimum. Even if h < bH, there remain the other two marginal benefits of greater deterrence:reduced enforcement expenditures and reduced chilling, both on account of there being fewerharmful acts to investigate.

C. Optimal Evidence Threshold

By analogy to expressions (4) and (5), the first-order condition for xT is

( ) ,15 H H H HB

HB B Bsf b h b e

d

dpsf b b

where

- 20 -

( ) .16 1d

dpP F b p sf b

B

HB B H H B H H H

The differences between this first-order condition and expression (5) largely parallel those givenfor the optimal level of enforcement effort. In expression (15), we have the additional “plus e”on the left side, indicating the additional advantage from deterrence of reducing enforcementexpenditures. Through expression (16), we have two additional effects that pertain to chilling. First, the direct chilling effect, reflected by PBN as in the original model (where the ratio ofeffects on harmful versus benign acts again depends on how the rate of mistaken sanctioningrises with the rate of correct sanctions as the evidence threshold is relaxed), is now weighted by1!FH(bH) because the mistaken imposition of sanctions can only happen when an investigation istriggered in the first instance. Relatedly, the second term in (16) indicates the marginalreduction in chilling effects on account of enhanced deterrence from the increase in pH thatresults from a reduction in xT. As with the interpretation of the first-order condition for optimalenforcement effort, we cannot really say that, because we have three more favorable factors, alower xT is optimal when enforcement is by investigation. However, we again see that the natureof the optimization problem differs qualitatively and in important respects that relate directly tothe concern with the chilling of benign acts.

One can also construct the analogue to expression (6), which compares raisingenforcement effort and lowering the evidence threshold in achieving a given level of deterrence,by combining the first-order condition for e, expressions (13) and (14), and that for xT,expressions (15) and (16), which yields

( ) .171

P

p

pf b b

p s

BB

H

B

H

B

H

B B B

H H

Note that the fractions πB/πH and πBN/πHN were both equal to one in model I.

Additionally, the first-order condition for xT may usefully be compared to the analogue toexpression (8) for the preponderance of evidence rule:

( ) .18 1 1

B

HB B BP F b

This expression differs from expression (8) for the monitoring technology in two ways. Mostapparent, the term 1!FH(bH) no longer appears on the left side because, with the investigationtechnology, it is known that a harmful act was committed. (Put another way, although theaggregate flow of individuals who commit harmful acts and end up before the tribunal has thefactor 1!FH(bH), this factor also appears in the aggregate flow of individuals who commit benignacts and end up before the tribunal, so the effects cancel.) Second, the fraction πB/πH on the right

- 21 -

side here simply equalled one under the assumptions for the monitoring technology. (The originof this fraction is that the base rate for those committing the harmful act – on the left side to theanalogue to expression (7), which is not produced here – has the factor πH , and the right side hasπB .) The differences between the preponderance of the evidence rule (18) and the first-ordercondition (15 and 16) are even more striking with the investigation technology. For example,the preponderance rule, as before, depends on the distribution function for benign acts whereasthe first-order condition again does not; however, the preponderance rule here does not dependon the distribution function for harmful acts whereas the first-order condition here does, bothreversed from the prior case. Let us now state:

Proposition 5: When enforcement is by investigation:a. The optimal evidence threshold, given implicitly by (15) and (16), generally differs

from the preponderance of the evidence rule, given by (18) (or any similar rule denominated interms of an ex post likelihood that an individual before the tribunal must have committed theharmful type of act).

b. Raising the evidence threshold xT may increase the frequency with which individualswho commit benign acts are mistakenly subject to sanctions.

For proposition 5.b, consider for this model the effect of raising the evidence threshold xT

on the frequency of mistaken applications of sanctions. For the monitoring model, the result wasindeterminate (recall proposition 4.c) because benign acts that are committed face a lowerfrequency of sanctions but the number of benign acts that are committed increases. In thepresent model, this net effect is multiplied by another factor, itself of indeterminate sign, whichjust equals dρB/dpH.28 The intuition is that, in the monitoring model, raising xT unambiguouslyreduces bB, the benefit level of the marginal chilled individual; here, whether that happens or theopposite depends on how ρB changes, which depends not only on the sign of PBN as before (whichis unambiguously positive), but also on additional terms. Specifically, examining expression(16), we can see two competing effects: on one hand, raising xT, which reduces pH and also pB,will reduce the frequency with which individuals who commit the benign act are sanctioned; onthe other hand, raising xT, which reduces pH, will reduce the deterrence of harmful acts, therebyresulting in more investigations and thus more opportunities for innocent individuals to bemistakenly subject to sanctions. Combining the above, we have the product of two terms ofindeterminate sign – the composite term from the monitoring case (summing the competinginframarginal and marginal effects from a change in the expected sanction on those committingbenign acts) and dρB/dpH – which is itself indeterminate in light of the fact that each depends onlargely separate factors. (The former, recall, has its sign indicated by whetherbBf B(bB) > 1!F B(bB), which can be contrasted with expression (16).) One might suppose that acommon case will be one in which the component common with the monitoring model will be

28The change in the expected number of mistaken applications of sanctions as one raises pH, which has asign opposite to that of raising xT, is given by

d

dpF b f b b

B

H

B B B B1

.

In the monitoring model, in place of dρB/dpH, one has πPBN.

- 22 -

such that the dominant effect involves the existing pool of unchilled individuals (perhaps mostindividuals commit their benign acts), with the change in the size of the pool being relativelyunimportant. For that case, in model I with monitoring, raising xT would indeed reduce thefrequency of false positives. However, in model II with investigation, it seems plausible that,especially when the evidence threshold is fairly high, deterrence effects regarding harmful actswould dominate so that tighter proof requirements would result in more benign acts beingmistakenly subject to sanctions.

5. Extensions

A. Auditing (Model III)

This subsection briefly describes a third model: enforcement by auditing. Regardingbehavior, the analysis is like that in model I. Regarding enforcement costs, there are similaritiesto model II: there, the pool of acts triggering costly scrutiny consists of harmful acts, whichlikewise trigger costly audits here. The difference here is that benign acts also lead toenforcement costs being incurred because they too are subject to audits.

To be explicit, modify model I as follows. The probability of scrutiny (here, an audit)will continue to be denoted by π, so as before we have bi / πpis for i = H, B. The enforcementvariable e will now be interpreted as the expenditure required per audit. (It is natural to considerthe case in which πN is constant, corresponding to costs per audit being invariant to the level ofaudits, but we will not restrict attention to that case for the sake of generality and to facilitatecomparability across cases.) Social welfare for the auditing model is as in expression (1), exceptthat e in the last term is weighted by the number of audits (the audit probability π times the sizeof the pool subject to audit, which is the total number of individuals who commit either type ofact). See expression (A10) in the appendix.

Note that, as with the monitoring model (but unlike with the investigation model), thestandard Becker (1968) argument applies. Note further that the new version of the Beckerargument introduced in section 3.C does not hold in this case: even though raising a nonmaximalsanction s while raising the evidence threshold xT so as to keep deterrence constant will, asbefore, reduce the chilling effect, which in and of itself is desirable, reduced chilling also meansthat more individuals commit benign acts, which increases the size of the audit pool, therebyraising enforcement costs (even though e and thus π are held constant).

The first-order conditions corresponding to expressions (3), (5), and (6) for this modelalso appear in the appendix (A11–A13). There are two sets of modifications. Regardingmarginal effects of the enforcement instruments on behavior, deterrence is now more valuable(making it possible that the optimum involves overdeterrence relative to first-best behavior) andchilling less costly, in both cases because fewer acts lead to fewer audits, which reducesenforcement expenditures. (Indeed, chilling is net desirable when bB < πe, that is, when thebenefit of the marginal benign act is less than the expected audit cost.) This pair of effectsmakes both greater enforcement effort and a lower evidence threshold more attractive. Inframarginally, greater enforcement expenditures (and thus a higher audit rate) raise costs for

- 23 -

those who commit either type of act, which makes raising enforcement less attractive. Oneshould, however, keep in mind the caveat presented in section 4’s analysis of the investigationmodel: that one cannot formally compare results across models because different technologiesand settings are envisioned. Finally, observe that because, as noted at the outset, behavioraleffects themselves are the same as in model I with monitoring, raising the evidence threshold xT

in this auditing model can increase the frequency of mistakenly sanctioning benign acts. In lightof the foregoing analysis – supplemented by that in section 3 on monitoring, where behavior isdetermined identically – we can state (by analogy to proposition 4) that:

Proposition 6: When enforcement is by auditing:a. The optimal evidence threshold, given implicitly by (A12), generally differs from the

preponderance of the evidence rule, given implicitly by (8) (or any similar rule denominated interms of an ex post likelihood that an individual before the tribunal must have committed theharmful type of act).

b. At the optimal evidence threshold, the minimum ex post likelihood that an individualmust have committed the harmful type of act in order to be subject to sanctions can take on any(admissible) value.

c. Raising the evidence threshold xT may increase the frequency with which individualswho commit benign acts are mistakenly subject to sanctions.

d. Raising the evidence threshold xT may result in a lower ex post likelihood that anindividual must have committed the harmful type of act in order to be subject to sanctions.

Before concluding this subsection, it is worth observing that the additional features ofthis auditing model are also pertinent in realistic settings with monitoring. For example, whenpolice patrols identify possible perpetrators, there may need to be follow-up investigation or atleast some sort of adjudication in order to generate and assess evidence, which obviously entailscosts, such as are included here. Specifically, the costs are triggered in all cases where themonitor ultimately picks out an act, which is likewise true with regard to the decision to conductan audit of some act. Thus, monitoring is also likely to be more costly when more acts (of bothtypes) are committed, giving rise to the additional effects here attributed to a model withauditing.

B. Socially Costly Sanctions

Suppose that sanctions, instead of being costless transfers, entail direct social costs.29 Notably, imprisonment is socially costly both because individuals’ utility loss is not offset by acorresponding gain and because the operation of prisons consumes resources. In particular,assume that each unit of the sanction s that is imposed entails a social cost of σ. The resultingexpression for social welfare (in the monitoring model) is:

29For prior analyses of socially costly sanctions (in models without errors), see Polinsky and Shavell (1984)and Kaplow (1990); and for discussion with error but in a single-decision framework like that in section 3.E, seeKaplow (1994). On risk aversion, see note 12.

- 24 -

( ) ( ) ( ) ( )

( ) ( ) ( ) .

( ) ( )

( ) ( )

19

W b h e p x s f b db

b e p x s f b db e

H T H

e p x s

B T B

e p x s

H T

B T

Compared to our original expression (1), each integrand now subtracts a term π(e)pi(xT)sσ, whichis the expected sanction for the type of act multiplied by the social cost per unit of sanctionimposed. This modification obviously makes the commission of both the harmful and the benigntype of act less socially desirable.30

The first-order conditions corresponding to expressions (3), (5), and (6) for, respectively,the optimal choices of e and xT and the combination indicating the relative desirability ofenforcement effort versus evidence threshold adjustment are as follows:

( )

.

20 1 1 1

1 1

p sf b h b p sf b b

s p F b p F b

H H H H B B B B

H H H B B B

( )

.

21 1 1

1 1

f b h b P f b b

F b P F b

H H H B B B B

H H B B B

( ) .22 1 11

P

p

pf b b F b

p sB

B

HB B B B B

H

Proposition 7: In the monitoring model, introducing or raising a social cost of sanctions hasambiguous effects on optimal enforcement effort, the optimal evidence threshold, and therelative desirability of increasing enforcement effort versus reducing the evidence threshold inachieving a given level of deterrence.

30Also of interest is that the traditional Becker (1968) argument for maximal sanctions holds because, sincethe product πs is held constant, sanction costs are also held constant. By contrast, the new variation presented insection 3.C need not hold because raising a nonmaximal sanction and increasing the evidence threshold in a way thatkeeps deterrence constant continues to have the benefit of reducing chilling costs but this reduction means that morebenign acts are committed and thus greater sanction costs will be incurred. Put more formally, expression (22) doesnot (as with expression (6)) guarantee that PBN > pB/pH.

- 25 -

To see why these ambiguous results follow, observe that, in each of these threeoptimality conditions, we have two new sets of effects. All conditions have an added term thatincludes the factors σ and 1!F i(bi). These indicate that both greater enforcement effort e and alower evidence threshold xT raise the expected sanction on individuals who continue to commitboth the harmful and the benign types of acts, which is now socially costly in itself. From thefinal terms on the right sides of expressions (20) and (21), this effect favors a lower e and ahigher xT, as one might have expected.

However, there is a second sort of effect regarding the marginal consequences ofdeterrence and chilling, indicated by the 1!σ before bH on the left sides of (20) and (21) andbefore bB on the right sides. As expected sanctions rise, more individuals with both types of actsare discouraged from acting, producing now a marginal social gain per individual who no longeracts equal to the expected sanction (which just equals the benefit of the respective marginalindividual) times the social cost per unit of sanction σ. This effect favors a higher e and a lowerxT, which may be contrary to what one would have anticipated (compare, for example, Lando2002, p. 606). Even the chilling of desirable acts now has a benefit: that fewer individuals aresubject to the socially costly sanction.

Because the first effect, changing the level of sanctions imposed on individuals whonevertheless act, is inframarginal, depending on the cumulative distribution functions, F i, andthe second effect just described is marginal, depending on the densities, f i, among other factors,costly sanctions have an ambiguous effect on optimal enforcement effort and the evidencethreshold. To confirm that either set of competing effects could be larger, observe that, at σ = 0,the terms with the F i vanish. Then, as we raise σ slightly from 0, the magnitude of theinframarginal effects can be small or large in a way that depends on parts of the distribution thatdo not affect the original optimum.31

Expression (22) indicates that, for the same reasons, the effect on the relative desirabilityof using enforcement effort versus a lower evidence threshold to achieve a given level ofdeterrence is ambiguous. On one hand, lower evidence thresholds (which are free)disproportionately target benign acts relative to greater enforcement effort (which is costly), andthis now has a further disadvantage of increasing sanction costs on individuals who are notchilled. On the other hand, there is a new advantage in that the net social marginal cost ofchilling benign acts is reduced on account of the resulting saving in sanction costs.

The foregoing analysis, particularly of expression (21) for the optimal evidencethreshold, thus upsets conventional wisdom that costly sanctions in the criminal context providea clear justification for the higher burden of proof employed there. One can also slightly modifythe foregoing extension to capture additional considerations. For example, another ex postconsequence of imprisonment and certain other sanctions (e.g., suspensions of permission toengage in activities, such as through license revocation) is to reduce the incidence of harmfulacts by making individuals who might otherwise commit them unable to do so for a period of

31Furthermore, in each expression, (20) and (21), each marginal term, bif i(bi), has precisely the same weightas its corresponding inframarginal term, 1!F i(bi).

- 26 -

time. Consider the case in which the reduction in social harm is proportional to the magnitude ofthe sanction, in which event the σ in the first integrand of expression (19) for social welfarewould be replaced with a lower value, σ!. (If the incapacitation gain were sufficiently great, σ!

would be negative.) The value of σ remains unaltered in the second term because incapacitationof individuals who commit benign acts is assumed not to reduce the commission of harmful acts. The result of this change would be to accordingly reduce the magnitudes of σ to σ! on the leftsides of expressions (20) and (21), and this lower σ! would multiply the first term in squarebrackets on the right sides (with σ continuing to weight the second term). We can immediatelysee that, as the consequence of these two sets of changes, the effects of an incapacitation benefiton optimal enforcement effort and on the optimal evidence threshold are ambiguous. Just asraising the social cost of sanctions has an ambiguous effect on how these instruments should beset, so does reducing this social cost, even to a negative value (a net social benefit) for just thosecommitting harmful acts.

Finally, although the analysis throughout this article has taken a standard welfaristapproach under which social welfare is determined by the benefits and harms of acts,enforcement costs, and now the costs of imposing sanctions, the present extension allows one aswell to characterize optimal policy if society cares per se about the correct and incorrectimposition of sanctions, as is commonly suggested (especially by noneconomists). Specifically,suppose that social welfare is higher by some amount per unit of correctly imposed sanction andlower by some other amount per unit of mistakenly imposed sanction (without regard tobehavioral effects, resource costs, and so forth). By simply altering the magnitudes of σ!and σ toreflect these considerations, the reinterpreted versions of expressions (20) through (22) wouldapply; in particular, the modified expression (21) would indicate the optimal evidence threshold. Yet again, the effects on optimal policy are ambiguous, contrary to the conventional view that,for example, associating a greater cost with mistakenly sanctioning the innocent favors a higherevidence threshold. Finally, if one cared only about such ex post effects, one could set σ! to anegative value equaling the social gain per unit of correctly imposed sanction, σ to the positivevalue for the social cost per unit of mistakenly imposed sanctions (regarding the signs, recall thatthe σ’s were originally presented as costs, not benefits), and remove the other terms relating tobehavior and enforcement costs. It is not suggested that this is a plausible social objective, andthe results have some peculiar implications,32 but this discussion indicates how the presentframework readily allows one to analyze a variety of objective functions.

32For example, under the posited ex post objective function, deterrence of harmful acts turns out to bedetrimental because society obtains its benefit from the correct imposition of sanctions less often, and chillingdesirable acts is beneficial because society less often suffers the welfare loss from incorrectly sanctioning benignacts. For criticism of nonwelfarist approaches, including that they can violate the Pareto principle, see Kaplow andShavell (2001, 2002). Some have suggested to me an objective function that only cares about the relative effects incases that actually come before the tribunal. Again, deterrence could be undesirable and chilling desirable becauseboth “worsen” the priors in that a lower fraction of cases before the tribunal involve true positives. Moreover, theoptimal overall enforcement scheme would involve virtually no prosecutions (and hence essentially no deterrence),bringing only the occasional case in which one was truly certain that the individual committed a harmful act. Also,welfare would improve by rounding up many innocent individuals and subsequently acquitting them, because theoutcomes ratio would improve. Accordingly, even by reference to objective functions that focus on ex posttreatment, it is not that easy to attempt to rationalize the existing legal system, including an ex post Bayesian burdenof proof concept.

- 27 -

C. Accuracy33

The accuracy of the decision-making process depends on the quality of the signal x,which in turn is indicated by the densities g i(x) or, equivalently, the distribution functions G i(x). There are two preliminary difficulties in analyzing the effects of the degree of accuracy on theoptimal level of enforcement effort and the optimal evidence threshold. First, in comparing twosignals, it may not be that one is unambiguously more or less accurate than another because therelative error rates may depend on the choice of the evidence threshold. Second, it is somewhatunclear what it means to hold the evidence threshold constant when comparing two tests. Forexample, one could hold xT fixed, but such would be incoherent if the two signals are distributedon different, non-overlapping intervals; more generally, the meaning of a given xT will differacross tests.

Accordingly, for concreteness let us consider a simple, special case where thecomparison is likely to be as clear as one might hope, in order to see what can be learned. Specifically, let the distribution functions be such that pi(xT) = , for x 0 [0,1], where( )1 xT i

αB > αH > 1. It is convenient to rewrite this as pB = (pH)α, where α = αB/αH. As discussedpreviously, the choice of xT can be understood equivalently as a choice of pH. This particularfunctional form relating pB to pH is convex (see section 2 for further discussion). With thisformulation, an increase in α represents an unambiguous increase in accuracy (except at 0 and1): for any given pH, pB falls. Analysis will be confined to choices where, at least initially, theevidence threshold xT under the higher α is set such that pH is higher and pB is lower.

Further restricting attention to the case of uniform distributions of benefits (constantdensities), reconsider the first-order conditions (3) and (5). When pH is higher and pB is lower,the marginal gain from deterrence falls (since bH is higher) and the marginal cost from chillingalso falls (since bB is lower), which have opposing effects on both the optimal level ofenforcement and the optimal evidence threshold. Regarding enforcement effort, there is anadditional set of effects: when pH is higher, the deterrence punch is greater (a given increase in ehas a larger effect in raising the expected sanction on harmful acts), and when pB is lower, thechilling punch is less (a greater increase in e has a smaller effect in raising the expected sanctionon benign acts), both of which favor greater enforcement effort. Regarding the optimal evidencethreshold, one might also wonder about the effect of increased accuracy on PBN. This willdepend on how pH and pB are chosen under the more accurate test, although the initial values andother parameters have implications for its magnitude (when it is chosen optimally). It mightseem natural to focus on the special case in which PBN remains the same (although this may beinconsistent with supposing that pH is higher and pB is lower).34

33Prior treatments of accuracy and law enforcement (in a model like that in section 3.E) include Kaplow andShavell (1994) and Kaplow (1994).

34Regarding condition (5) for the optimal evidence threshold, consider first the case where the initiallyoptimal threshold is high, implying a low pH. With greater accuracy, pB will rise slowly, so chilling costs will besmall, and one will not reach a given level of PBN until pH is higher, indicating the optimality of a lower evidencethreshold (subject to the usual qualifications depending on the f i’s). However, if the optimum is initially far enoughto the right (a low evidence threshold), a more accurate signal could have the opposite effect, favoring a lower pH,

- 28 -

It is also instructive to revisit the comparative first-order condition (6) that indicates therelative virtues of lowering the evidence threshold compared to raising enforcement effort inachieving a given level of deterrence. Again, even assuming a constant density function, wehave competing effects. The reduction in pB implies, as mentioned, a lower bB, a reducedmarginal social loss from chilling the marginal individual, which reduces the relativedisadvantage of lowering the evidence threshold (which, recall, involves relatively more increasein the marginal chilling punch in achieving a given level of deterrence than does raisingenforcement effort.) Other factors, by contrast, make the use of enforcement effort relativelymore favorable. The reduction in pB and increase in pH both reduce the fraction pB/pH: onaccount of higher accuracy, increased enforcement effort now has less chilling punch and moredeterrence punch than before, as noted; i.e., enforcement effort’s relative benefit in terms of thechilling–deterrence tradeoff is enhanced. The pH in the denominator on the right side indicatesthat greater accuracy makes it less expensive to achieve a given level of deterrence augmentationthrough increased enforcement effort; i.e., enforcement effort’s resource cost disadvantage isreduced. And, again, we have PBN, about which little in general may be said. Considering aswell that we are limiting analysis to the case of a constant density function and a particularformulation of the signal and how it changes with increased accuracy, what can be concludedabout the effect of accuracy on the optimal setting of other enforcement instruments is limited.

Finally, it is interesting to inquire into the value of accuracy in the present context andhow optimal accuracy is determined. In our special case in which pB = (pH)α, and supposing thatall other instruments are set optimally, the marginal value of accuracy can be determined bytaking the derivative of social welfare in expression (1) with respect to α, while holding pH fixed. (By the envelope theorem, we can ignore as second-order further influences on welfare due tothe optimal adjustment of pH, which was a focus of the preceding discussion.) This marginalvalue, which involves reduced chilling costs, can be expressed as ![pB ln pH]πsγf B(bB)bB, whichis fairly similar to the right side of expression (4). The terms after the brackets are the product ofan expected sanction effect, πs, the (population-mass-weighted) density for the benign act,γf B(bB), and the benefit of the marginal act no longer chilled, bB. The term in brackets is thederivative of pB with respect to α for the posited functional form, which is negative because[ln pH] is negative (in light of the fact that pH < 1). Not surprisingly, the value of enhancedaccuracy is greater the larger is the drop in pB and the greater is the marginal chilling cost. Regarding the cost of accuracy, suppose that the enforcement expenditure term e in expression(1) is replaced by ec(α), where cN(α) > 0. At the optimal level of accuracy, then, the foregoingexpression for the marginal value of accuracy would be equated to ecN.

D. Relative Concentration of Chilling Effects

Model I assumes that harmful acts and benign acts are scrutinized at a common rate π,with γ indicating the mass of opportunities to commit benign acts (with that for harmful actsnormalized to one). Consider now the possibility that, relative to this benchmark, the chillingpotential of law enforcement is more concentrated (or dispersed) by introducing a factor λ thatmultiplies π for benign acts and also divides γ: λ greater (less) than one indicates more highly

that is, a higher evidence threshold.

- 29 -

concentrated (dispersed) chilling effects. (If λ = 2, only half as many individuals haveopportunities to commit benign acts that may become subject to scrutiny, but such acts arescrutinized with twice the frequency.) The expression for social welfare becomes

( ) ( ) ( ) ( ) .23

W b h f b db bf b db eH

p s

B

p sH B

In the analogues to the first-order conditions (3) and (5) for the optimal level ofenforcement effort e and the optimal evidence threshold xT, as well as condition (6) indicatingthe optimal relative use of the two instruments to achieve a given level of deterrence, the λ’sfrom the second term of expression (28) cancel (we have 1/λ times as many potential benign actsand λ times as many are subject to sanctions, ceteris paribus), so all the expressions are the sameas before. Nevertheless, the results differ because we now have bB = λπpBs. A higher λ (moreconcentrated chilling effects) means that the social cost of chilling the marginal act is higher:those individuals whose benign acts are potentially subject to mistaken sanctions face a higherexpected sanction, so the marginal type has higher benefits, raising the marginal social cost. (Byanalogy, it is familiar that applying a tax to half as many people but at twice the rate causes moremarginal and total distortion; the prospect of being fined in this model is tantamount to beingsubject to a probabilistic tax on the activity.) Just as with propositions 1b and 2b givingcomparative statics for raising γ (indicating a relatively greater importance of chilling effects), soa higher λ on this account tends to make optimal a lower level of enforcement effort and a higherevidence threshold, and also a greater relative reliance on enforcement effort than on a lowerevidence threshold to achieve a given level of deterrence.

This characterization is incomplete, however, because in each of our three optimalityconditions, the term bB is multiplied by f B(bB): there is a reinforcing or countervailing effectdepending on whether f BN(bB) is positive or negative. The originally stated sign of the net effectcontinues to govern (for a marginal change in λ) if and only if f B/bB > !f BN(bB) (which obviouslyholds when f BN(bB) $ 0).

6. Conclusion

Setting the evidence threshold or burden of proof is an important and often overlookedpolicy instrument, along with the traditionally studied tools of enforcement effort and sanctions. Incorporating it into a framework that allows for errors not only to reduce deterrence but also tochill desirable behavior casts additional and sometimes quite different light on previouslyexamined questions and presents new and challenging ones. Further interest is added by the factthat there are important, qualitative differences across the enforcement technologies ofmonitoring, investigation, and auditing. Among the distinctions are that, when enforcement isby investigation, optimal sanctions need not be maximal, there need not be underdeterrence atthe optimal level of enforcement, and raising the evidence threshold may raise the frequencywith which individuals who commit benign acts are mistakenly sanctioned but possibly underopposite circumstances compared to when enforcement is by monitoring or auditing.

- 30 -

Characterizations are offered for the optimal level of enforcement effort and for theoptimal evidence threshold in all three models. It is explained how these two instruments aresubstitutes in achieving the deterrence of harmful acts. The key differences are that raising effortinvolves resource costs, unlike lowering the evidence threshold, but the latter tends to berelatively less discriminating regarding benign acts and thus entails greater chilling costs (at leastin the neighborhood of the optimum). Briefer attention is devoted to the often simpler questionof optimal sanctions, including formulation of a new sort of Becker (1968) argument – favoringraising sanctions while raising the evidence threshold, which can keep deterrence andenforcement costs fixed while reducing chilling effects. This new argument for maximalsanctions holds in some cases in which the traditional one does not, thereby upending importantresults in prior literature.

Attention is also devoted to the relationship between the setting of an evidence threshold– a minimal level of evidence (value of a signal) required to impose sanctions, the direct policylever analyzed throughout – and conventional understandings about the burden of proof. Thelatter, embodied for example in the preponderance of the evidence rule that requires it to be morelikely than not that the individual before the tribunal committed the harmful act, is an ex postBayesian notion that has very little connection to the first-order condition that characterizes theoptimal evidence threshold in any of the three models. One implication is that, at the optimalevidence threshold, the requisite ex post likelihood that an individual must have committed theharmful type of act in order to be subject to sanctions can take on any value. Also, raising theevidence threshold may increase the frequency with which individuals who commit benign actsare mistakenly sanctioned and may reduce the ex post likelihood that an individual before atribunal must have committed the harmful type of act in order to be sanctioned. Prior work, byfocusing on a much simpler case (with only one type of act and thus no possibility of chillingeffects), has tended to obscure the magnitude of the gulf between the different approaches to theburden of proof.

An extension considers costly sanctions and finds that the effects are ambiguous; forexample, raising the evidence threshold, which may seem attractive because, conditional onbehavior, fewer individuals who have committed both harmful and benign acts are sanctioned,has the offsetting effect that more of both types of acts are committed ex ante, thereby expandingthe pool who might receive the costly sanctions. Incapacitation is introduced; in the caseconsidered, it is tantamount to reducing (possibly to a negative level) the social cost of sanctionsas applied to actually harmful acts. Social benefits or costs associated with the correct orincorrect imposition of sanctions, viewed as ends in themselves, are analyzed similarly. Enhanced accuracy has ambiguous effects on the optimal setting of enforcement instruments. And, when chilling effects are more concentrated, their social consequences tend to be worse(but not necessarily so), in which case less enforcement effort and a higher evidence thresholdare optimal and relatively greater reliance on the former instrument is desirable.

Even though three different enforcement technologies are analyzed, much remains to be

- 31 -

done to provide a comprehensive view of evidence thresholds (or proof burdens).35 One of themost significant unexamined dimensions concerns enforcers’ behavior in gathering evidence andbringing cases, regarding both public officials, such as government prosecutors or those inadministrative agencies, and private litigants. Incorporating this feature – aspects of which havebeen examined in basic settings, often with exogenous behavior and stipulated, constant socialcosts to errors (see the references in note 2; Prendergast (2007) on optimal biases of governmentofficials; and Kamenica and Gentzkow (2011), who more generally investigate persuasion) –renders endogenous both enforcement levels and costs, making the burden of proof an even moremomentous and more difficult to analyze enforcement instrument. For informal discussion,including greater attention to the application to civil litigation, see Kaplow (2012a). Anotherpossibility is that the legal system might impose one sort of proof burden at an initial stage but adifferent one after further examination.36 Such a distinction is involved in prosecutors’ oragencies’ screening processes and in many systems of adjudication that eliminate cases at earlystages if what can be established at that point fails to meet some minimum standard.

The present inquiry, although preliminary in significant respects, helps illuminate theseand other questions. It provides characterizations of the optimal use of different enforcementinstruments, including the burden of proof, and shows how they interact – how best to achieve atarget level of deterrence through adjusting multiple instruments. It demonstrates the largedifference in the analysis when errors may chill desirable behavior, a phenomenon absent in pastwork. It exposes how results change qualitatively when enforcement is by investigation or byauditing rather than by monitoring. And it presents the stark differences between determinantsof an optimal evidence threshold and of conventional notions of the burden of proof, most ofwhich do not arise in simpler models.

35For further extensions and for relation of the analysis to legal perspectives on the burden of proof, seeKaplow (2012a, 2012b).

36This extension is the subject of work in progress, tentatively entitled “Optimal Multistage Adjudication.”

- 32 -

Appendix

Proof of Proposition 1. To begin, it is helpful to state the second-order condition for aninterior optimum for enforcement effort.

( )

,

Ad W

dep s f p sf p s f h b

p sf b p s f b p s f

H H H H H H H

B B B B B B B B

1

0

2

22 2 2 2 2 2

2 2 2 2 2 2

where the arguments of the f i (each evaluated for the pertinent marginal benefit level bi) aresuppressed for convenience.

To prove part a, we take the derivative of the first-order condition (2) or (3) with respectto h and solve for de/dh, yielding the following:37

( ) .Ade

dh

p sf

d Wde

H H

2 02

2

Moreover, the numerator, reflecting the rise in deterrence of harmful acts, accords with theintuition given for why raising h raises e*.

To prove part b, we examine de/dγ:

( ) .Ade

d

p sf b

d Wde

B B B

3 02

2

Again, the result accords with the stated intuition, as the numerator indicates the marginal lossfrom deterring the marginal benign act.

37Although it is not in general sufficient to determine comparative statics by differentiating the first-ordercondition in problems in which the second-order condition does not hold globally (which is true here), examinationof the first-order condition (2) or (3), moving all the terms to the left side (which gives the expression for dW/de)makes apparent that there is no concern. Specifically, for part a, it is obvious that raising h raises the value of dW/dethroughout, so if there were two local optima that were each a global optimum, it is clear that raising h would makethat associated with a higher e the unique global optimum. Likewise, for part b, raising γ has the opposite effect. Analogous reasoning is applicable with regard to all parts of proposition 2.

- 33 -

Establishing part c is more complicated. Solving for de/ds yields:

( ) .A

de

ds

p f p sf h b p sf

p f b p sf b p sf

d Wde

H H H H H H H

B B B B B B B B

4

2 2

2 2

2

2

It is easy to assert that expression (A4) cannot unambiguously be signed, but it requires furthereffort to confirm that this is so. It suffices to construct examples consistent with the model’sassumptions that yield opposite signs for the expression.

We will confine attention to the numerator and consider the case where both of thedensities f i are constant on the unit interval and γ = 1. Furthermore, we can use the first-ordercondition (3) to substitute for the leading terms in each row of the numerator. When all this isdone, the numerator simplifies to

( ) .As

s p pH B51 2 2

It appears that a low value of s renders expression (A5) negative and a high value makes itpositive. However, since the values of π and πN are determined by the optimal choice of e, whichin turn depends on s, some further analysis is required.

Suppose that s initially equals 0.5. We also know that π cannot exceed one and that the pi

cannot exceed one (nor can their squares). Hence, to obtain a negative value for expression (A5)and thus a positive value for (A4), it is sufficient that πN < 2. But we can choose a π(e) functionand an h that is high enough to assure this at the optimal level of e.

For the other possibility, first observe that, for any burden of proof, the value of the termin parentheses will equal some constant. Thus, we can guarantee that expression (A5) is positiveand hence (A4) negative if ππN is sufficiently large (at some given initial value of s). Supposethat the function π is such that it obtains a value of 0.5 essentially for free (we can choose thefunction such that the level of e such that π(e) = 0.5 is arbitrarily small), which places a floor onthe value of π. We are left with demonstrating that it is possible that πN is sufficiently large giveneach of these other factors that can be treated as constants (or floors). Again, this can beaccomplished with an appropriate function π(e) and value of h. Specifically, choose h highenough that it is beneficial to raise π to at least 0.5, but not so high to drive πN below the desiredvalue. (Note that we have essentially chosen a π function such that πN is arbitrarily large untilπ = 0.5 and then drops sufficiently rapidly since we know π cannot exceed 1. If h was chosen sothat, at π = 0.5, one was indifferent to a free increase in π, one would increase π no further andthus the optimal πN can be made as high as we like.)

Proof of Proposition 2. As discussed in section 2, it is equivalent to view the problem of

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choosing the optimal xT as instead one of choosing the optimal pH. The latter course will beadopted here because it is somewhat less cumbersome notationally. Obviously, all comparativestatics in terms of pH will, because dpi(xT)/dxT = !gi(xT), have the opposite sign of those for xT,the subject of the proposition .

Accordingly, begin with the second-order condition for an interior optimum for pH. (Throughout, the version of the first-order condition in expression (5), which cancels πs fromeach side, is employed.)

( )

.

Ad W

dpsf sf h b

P f b P sf b P sf

H

H H H

B B B B B B B B

6

0

2

2

2 2

To prove part a, we take the derivative of the first-order condition (4) or (5) with respectto h and solve for dpH/dh.38

( ) .Adp

dh

s

d Wdp

H

H

7 02

2

As explained, raising h makes the marginal deterrence benefit greater to that extent, producing ahigher pH* and thus a lower xT*.

For part b, we examine dpH/dγ:

( ) .Adp

d

P f b

d Wdp

H B B B

H

8 02

2

Because there are more benign acts, the marginal chilling cost of a higher pH is greater, so pH* islower and thus xT* is higher.

For part c, we can solve for dpH/ds:

38Regarding possible multiple optima, see note 36.

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( ) .A

dp

ds

p f p f h b p P f b p P f

d Wds

HH H H H H B B B B B B B

92

2

If the densities f i are constant (uniform distributions), the middle terms in the numerator, whichotherwise could have either sign, equal zero. The first and last terms are positive and thedenominator is negative, so under the stated assumption, dpH/ds < 0. The first term indicates theincrease in the marginal forgone benefit from harmful acts times the deterrence punch (recall thatπs has been cancelled from each side of the first-order condition before performing thisderivation). Likewise, the final term is the increase in the forgone benefit from benign acts timesthe chilling punch. Since both forgone benefits are larger when s is higher, pH* is lower and thusxT* is higher.39

Auditing (Model III). The social welfare function is:

( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) .

( ) ( ) ( ) ( )

A W b h f b db bf b db

e F e p x s F e p x s e

H

e p x s

B

e p x s

H H T B B T

H T B T

10

1 1

As suggested in the text, the first two terms are the same as those in expression (1) for themonitoring model, whereas the last weights e by the number of audits, which in turn equals theaudit probability π times the size of the pool subject to audit, that is, the total number ofindividuals who commit either type of act.

The first-order conditions corresponding to expressions (3), (5), and (6) for this modelare as follows:

39Further exploration of (A9) can take advantage of the fact that the numerator is similar to the second-ordercondition (A6), particularly when the latter is multiplied by !pH/s. The differences are that (A6) has an additionalterm (the middle one, with PBO), and the last two terms of (A6) (after the modification) have pHPBN whereas (A9) haspB – the former exceeding the latter at an optimum, as discussed in section 3.C. Because modified expression (A6)’sadditional term is positive and the total of the modified expression is positive (we did multiply through by negativeone), it is natural to consider a case where, at the optimum, the second-order condition is barely satisfied, suggestingthat the numerator of (A9) might therefore be negative, with the result that we would have dpH/ds > 0. However, thispossibility is not so easy to establish because the relative magnitude of that term in (A6) is affected (among otherthings) by the magnitude of PBO, the value of which, at the optimum, depends on the level of pH, which is alsorelated to the difference between pHPBN and pB. Accordingly, for the present it is left as a conjecture that this othercase can arise.

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( )

.

A p sf b h b e p sf b e

e F b F b

H H H H B B B

H H B B

11

1 1

( ) .A f b h b e P f b b eH H H B B B B12

( ) .A Pp

pf b b e

e F b F b

p sB

B

HB B B

H H B B

H13

1 1

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